Download Statistical Mechanics of a Time-Homogeneous System of Money

Statistical Mechanics of a Time-Homogeneous System of Money
and Antimoney
Andreas Schacker, Matthias Schmitt, and Dieter Braun
Ludwig-Maximilians-Universität München
(Dated: September 27, 2012)
Abstract
Financial crises appear throughout human history. Their interplay with the real economy has
puzzled generations of economists and is the subject of ongoing research. There are many schools
of thought on as to what are the actual causes of such crises and how monetary policy can try to
avert them. In particular, it has been suggested that the creation of credit money might be a source
of financial instability.
We explain how the credit mechanism in a system of fractional reserve banking leads to non-local
transfers of purchasing power which also affect non-involved agents. In an attempt to overcome this
issue and motivated by an analogy to physics, we impose the local symmetry of time homogeneity
on the monetary system. The resulting full reserve banking system suppresses non-local transfers
upon credit creation but has often been accused of stalling bank lending.
Starting from this premise, we consider a bi-currency system of non-bank assets (’money’) and
bank assets (’antimoney’) in which a payment can either be made by passing on money or by
receiving antimoney. As a result, a free floating exchange rate between non-bank assets and bank
assets is established. Interestingly, credit creation is then replaced by the simultaneous transfer
of money and antimoney at a negotiated exchange rate. By introducing this novel mechanism
of liquidity transfer, the issue of credit crunches commonly associated with full reserve banking
systems might be mitigated. Tools from statistical physics promise to provide new insight into
monetary and creditary dynamics.
PACS numbers: 89.65.Gh
Keywords: Statistical Mechanics of Money, Credit Theory, Antimoney, Time-Homogeneous Monetary System, Liquidity Market, Inflation, Random Economy, Price Development, Locality in Money Transfers,
Wealth Distribution, Bookkeeping Mechanics
1
I.
OVERVIEW
We will first present the idea in a nutshell. In the following, we analyze a monetary system
of money and antimoney that has only recently become possible with the advent of mobile
computers and cryptography. One of the motivations are the non-local effects of credit
creation. Once a payment is performed by the creation of credit, the onsetting inflation
affects all other market participants. Credit holders profit from the monetary inflation while
money holders suffer from the inflation. As a result, the decision between two credit partners
affects all other market participants without a counteracting mechanism.
The result is a moral hazard. Products are bought by credit creation at price levels
before the inflationary effects and later can be sold at higher price levels after the market
equilibration of the inflationary pulse. This phenomenon can be seen in the contemporary
credit crisis. Everyone suffers from mediocre credit creation of a small number of banks. It
is likely that the moral hazard leads to an overboarding credit creation and eventually to
periodic debt defaults.
The following monetary structure is borrowed from the physics of energy and momentum
conservation [1] and is inspired by the Noether theorems of time homogeneity. Instead of
relying on interest rates to judge the future value of past investments at the credit creation,
a local exchange rate compares values of the past (debt = antimoney) with values of the
future (money). This judgement is performed globally and in real time by all participants in
any monetary transaction, in stark contrast to interest rate determinations which are judged
locally and for finite time spans between the credit creating partners.
The differences from the existing monetary system are rather subtle. Money and antimoney units are normalized to the number of subjects in an economic space. This means
both money and antimoney are issued upon entering and collected when leaving the economic space. Importantly, money and credit are issued in differing currencies and their
mutual value is judged by an exchange rate.
Such a differentiation would already be possible today: money consists of bank liabilities
and non-bank assets while antimoney is memorized as bank assets and non-bank liabilities.
Both units are never added or subtracted during monetary transactions and are structurally
a money currency and a debt currency [2].
Instead of creating credit by new money and debt units, liquidity is transferred by the
2
concurrent transfer of money and antimoney. The exchange rate between both plays the role
of interest rates in the contemporary system. If investments of the past were good, money
prices drop (enhancing the value of money units) and debt prices increase (making it easier
to get rid of debt units accumulated by the past investment). Credit is judged in real time
and liquidity is traded at a local level.
Clearly, an electronic system has to guarantee that the antimoney units do not vanish,
something that was not possible without electronic payment systems. Also, similar to contemporary structures of inheriting, money insurance schemes have to distribute the risk of
subjects leaving the economic space. In this case the same money and antimoney units
previously issued upon entering have to be collected.
Lacking the ability to conduct economic experiments, we first study the economic structure under the assumption of random transfers. Under random transfer, the stability of
economic structures can be crucially tested as the random walk tends to create inflationary
scenarios in creditary monetary systems [2].
II.
BACKGROUND
The origin of money and even a correct definition of money are still subject to debate
[3, 4]. Most economic textbooks today define money by its functions. If something serves
as a means of exchange, as a unit of account and as a store of value, it is money (“Money is
what money does.” [5]). For a deeper analysis, it is useful to distinguish between commodity
money and representative money. Commodity money fulfils the aforementioned criteria
because of a perceived intrinsic value of the material it is made of, such as gold or silver.
Representative money on the other hand is essentially a claim on goods or services to be
rendered in the future and thus the ability to serve as money is heavily dependent on
social and legal measures to enforce these claims. It is also clear that representative money,
being a social contract, is independent of its physical representation. This becomes most
apparent in electronic payment systems. Historically, the creation of representative money
is inextricably linked to the record of debt relations [6, 7]. If a buyer wasn’t able to pay
for a good immediately, the seller could grant him a credit if he had sufficient trust in the
economic potential of the buyer. Both transaction partners would record their debt relation,
e.g. by the use of tally sticks or clay tablets [8] (see Figure 1). If the debtor was a highly
3
(a) A Sumerian clay tablet (b) Tally sticks
FIG. 1. Two examples for records of debt relations which served as currencies. Tally sticks were
commonly used in medieval Europe. After marking a wooden stick with notches that represented
the transactional value, the stock was split into halves of different lengths. The creditor kept the
longer part which was called stock and the debtor was given the shorter one, called foil. As the stock
represented a claim to future income, they were actively traded. The term stock market derives
from this practice. Note that in contrast to commodity money, the creation of representative money
comes at virtually no cost.
visible agent with a good economic standing, the claims drawn on him could themselves be
used in trades with other market participants. They thus served as a currency. In principle,
everyone could issue his own currency by handing out promissory notes [9]. Of course,
incomplete information about the market participants and the overhead of enforcing the
claims prohibited wide circulation of these notes.
Once the issuer of promissory notes has reached a certain level of trustworthiness and the
notes begin to circulate as means of payment, it is often not necessary for the issuer to be
able to meet all his obligations at all times [10]. It suffices if he meets any demands that are
actually made. Out of convenience, most of the creditors will then continue to use the notes
in trades as they are confident that they could cash the notes at all times if they so desired.
As long as this trust is maintained, the issuer can now create new money by accommodating
loans: he can hand out additional promissory notes to a borrower who has now acquired
liquid means of payment and who in return is obliged to pay back the loan after a certain
time, usually with an interest rate.
This service is nowadays provided by commercial banks. The money in a commercial bank
4
account is legally a claim on central bank money which in most countries is fiat money, i.e.
money that has been declared legal tender by government decree. The majority of financial
transactions today is carried out by commercial bank money. As the commercial bank will
usually meet its liabilities and hand out central bank money to the customer on demand, the
trust into the solvency of the commercial bank guarantees the acceptance of its “promissory
notes” (i.e. a client’s demand deposit account) in the economy. However, if this trust is
lacking, bank runs might occur and the bank will have problems finding creditors on the
interbank lending market. The maximum amount of money that a commercial bank can
create via loans is usually limited by fractional reserve banking. One problem with the
creation of money via loans is the possible devaluation of the currency [11]. Moreover,
as discussed later, credit creation creates non-local transfers of purchasing power which
adversely affect net asset holders. Starting from this observation, we will propose a monetary
system of money and antimoney that remedies this deficiency and introduce a novel transfer
mechanism for obtaining liquidity.
III.
CREDIT CREATES NON-LOCAL TRANSFERS OF PURCHASING POWER
Consider a simplified economy of N + 1 agents where the (N + 1)th agent has gathered
sufficient trust in that the other N agents are willing to accept its promissory notes as means
of payments (see section II). In the following we will call this agent “bank”. For simplicity,
we assume that the bank’s promissory notes are the only currency that is in circulation
within the economy. Note that for our analysis we can use the terms “promissory notes” and
“demand deposits” interchangeably. Both represent a client’s claim to an underlying entity
that the bank promises to meet.
Every agent keeps track of his monetary wealth by a so-called T-account in which his
assets are recorded in the left column and his liabilities in the right column (cf. Figure 2).
Note that the agents’ assets are liabilities of the bank and vice versa. The amount of assets
(n)
(n)
and liabilities that agent k holds at time tn are denoted ak and lk , respectively. The total
monetary supply at time tn is
M (n) =
N
X
(n)
ak =
k=1
N
X
(n)
lk .
(1)
k=1
Bookkeeping shall be implemented in a fail-safe electronic way such that no monetary units
5
FIG. 2. All monetary transactions can be visualized in a T-account. Here, only the updates in the
balance sheets are shown. Figure (a) shows a direct transfer of asset units from buyer i to seller j.
A payment via credit creation (b) leads to an increased money supply and – as will be shown later
– to non-local transfers of purchasing power. The reverse process of credit annihilation (c) occurs
when a creditor pays a debtor and leads to a decrease of the total money supply. Note, of course,
that credit annihilation can only occur after credit creation.
are lost or destroyed. If agent i buys some good from agent j and they have agreed on a
price ∆ > 0, then agent i has several payment options. He can directly transfer the amount
∆ from his asset account to j’s asset account. Given that the trade happened between t1
and t2 and no other transactions occurred in that time period, the asset holdings after the
trade will be
(2)
(1)
(2)
(1)
(2)
(1)
ai = ai − ∆
aj = aj + ∆
(2)
ak = ak ,
where k 6= i, j are the agents that are not involved in the trade. The liabilities are not
affected in this case. Figure 2 (a) shows the corresponding bookkeeping operation on the Taccounts. As can easily be seen, the total money supply hasn’t changed: M (2) = M (1) ≡ M .
If we look at the relative monetary wealth that each agent holds of the total monetary supply
(n)
(n)
wk =
(n)
ak − lk
,
M (n)
6
(3)
we see that the relative monetary wealth of the agents which are not involved in the trade
remains unaffected:
(2)
(2)
wi
(2)
wj
(2)
wk
(1)
(2)
(1)
a − li − ∆
ai − li
(1)
= i
< wi
M
M
(2)
(2)
(1)
(1)
aj − lj
aj − lj + ∆
(1)
=
=
> wj
M
M
(2)
(1)
(2)
(1)
ak − lk
ak − lk
(1)
=
=
= wk .
M
M
=
(4)
This is not the case if i chooses to pay by credit creation (cf. Figure 2 (b)). In that case,
the relevant changes of the agents’ accounts are
(2)
= li + ∆
(2)
(1)
(2)
(1)
li
(1)
aj = aj + ∆
(5)
ak = ak .
But now, the total money supply that can be used in trades has increased: M (2) = M (1) +∆.
If we now look at
(2)
(2)
wi
(2)
wj
(2)
wk
(1)
(2)
(1)
a −l −∆
a −l
(1)
< wi
= i (2) i = i (1) i
M
M +∆
(2)
(2)
(1)
(1)
aj − lj
aj − lj + ∆
(1)
=
=
> wj
M (2)
M (1) + ∆
(2)
(2)
(1)
(1)
ak − lk
ak − lk
(1)
=
= (1)
6= wk ,
(2)
M
M +∆
(6)
we see that the relative monetary wealth of non-involved agents k after the trade has changed.
The shift in relative asset and liabilities holdings is given by
(2)
(1)
∆wk = wk − wk
(2)
(2)
(1)
(1)
ak − lk
ak − lk
−
M (2)
M (1)
(1)
(1)
(1)
(1)
ak − lk
ak − lk
= (1)
−
M +∆
M (1)
∆
(1)
(1)
= − ak − lk
.
(1)
M · (M (1) + ∆)
=
(7)
Thus, we see that credit creation is potentially beneficial for debtors and adversely affects
creditors (see Figure 3). Only in the case of ak = lk , non-involved agents are unaffected by
P
credit creation. Indeed, as
∆w = 0, we see that there is an indirect transfer of relative
7
FIG. 3. The shift ∆wk of relative asset and liability holdings that non-involved agents experience
when a trade is executed either by direct asset transfer or by credit creation. Note that in the case
of a direct transfer, non-involved agents experience no change at all. In the case of credit creation,
the agents’ relative share of assets and liabilities decreases and thus agents with ak > lk are at a
disadvantage according to equation 7. Conversely, if ak < lk , agents profit from credit creation.
It is only when agents hold the same amount of assets and liabilities, that they are unaffected by
credit creation.
monetary wealth from creditors to debtors. In the case of credit annihilation (see Figure 2
(c)), the reverse effect occurs. As the total money supply decreases, the sign in equation
7 is reversed. One might conjecture that both effects somehow cancel. It has been shown
that unrestricted fractional reserve banking leads to an increase of the money supply without
bounds [2]. This is due to the fact that the number of loans (which create asset-liability pairs)
cannot be negative. In reality, bank lending is not completely unrestricted but nevertheless
a net increase of credit is confirmed by virtually all statistics about the money supply M1
(see e.g. [12]).
Let us now assume that agent k knows that the trade between i and j happened by credit
creation and that the transferred amount of money was ∆. If k previously charged a price
∆(0) for his services, he might now set a new price ∆(2) in order to compensate for his loss
of relative monetary wealth. Hence, from the condition
ak − lk + ∆(2)
ak − lk + ∆(0)
=
,
M +∆
M
8
(8)
we find that the expected new price should be
(2) ∆
∆
(0)
1+
∆
= (ak − lk )
+∆
M
M
∆
,
= ∆(0) 1 +
M
which is obviously larger than ∆(0) . (Note that hak i = hlk i =
number of market participants.) Moreover, as
∆(2)
∆(0)
M
N
(9)
with N being the total
= f (ak , lk ), there is no global factor by
which agents k could rescale their prices in order to get back to the initial situation. Of
course, this reaction of agents to inflation expectations assumes a widespread knowledge
about credit creation processes. It also assumes that agents raise their prices just enough to
hold the same fraction of the total money supply before and after the creation. Within the
scope of above discussion, we see that credit creation – ceteris paribus – leads to increasing
price levels. Note that in a real economy, those agents who are “closest” to the credit creation
profit the most from it as they will still face the old prices. This effect has already been
described by Richard Cantillon in the 18th century and has subsequently been named the
Cantillon effect [13].
Of course, credit creation is not the only source of inflation. Inflation - especially in the
short run - depends on several factors including but not limited to money supply, technological advancement, real demand, production and stability of governments. Most economists
believe an extension of the money supply to be the prime cause for inflation in the long term
[14].
The question of the particular benefits and disadvantages of inflation is an ongoing debate.
In most countries, general price stability has become one of the major goals of monetary and
fiscal policies [15–17] and in some even the most predominant one [17]. Inflation, whether
it be anticipated or unanticipated, is costly for society due to deadweight losses [18]. For
small inflation rates (around 2%), most economists have come to consider the negative
consequences being outweighed by a range of positive effects [19–22]. These include an
increased maneuverability of central banks near the zero lower bound of nominal interest
rates and the ability for firms to counteract the downward rigidity of nominal wages. As
deflation is often considered even worse than inflation, a small rate of inflation can also
serve as a buffer. However, the economic profession is still divided over the magnitude of
these effects [23]. Recent research has indicated that the effect of rigid wages might have
been overestimated in the past [24]. A recent working paper of the International Monetary
9
Fund also suggests that the corridor of acceptable inflation rates might be narrower than
previously thought [25].
The issue of money-induced inflation has of course been recognized and debated, e.g.
in connection with the bank charter act of 1844 which prohibited commercial banks from
issuing promissory notes [10] and by several other authors [11, 26–31].
In order to prevent non-local transfers upon credit creation and to attenuate the Cantillon
effect, the usually proposed solutions essentially restrict credit creation by commercial banks
or prohibit it entirely [32–35]. These proposals have come to be known as narrow banking,
full reserve banking or 100% reserve banking. Full reserve banking has long been dismissed
by mainstream economists, but in the view of the current financial crises, the idea has been
gaining some traction again lately [36, 37]. The main criticism of full reserve banking is its
alleged inability to provide an economy with the necessary liquidity, i.e. even agents whose
creditworthiness is ensured, cannot easily obtain means of payment for future investments
[38, 39].
In the following section we propose a full reserve, bi-currency system which might be able
to close that gap.
IV.
A TIME-HOMOGENEOUS BI-CURRENCY SYSTEM OF MONEY AND AN-
TIMONEY
We have seen (Figure 2) that representative money always consists of an asset-liability
pair: the asset part is the actual claim to a good or a service to be rendered in the future
and the liability part is the obligation of the debtor to provide it. The main idea for a
symmetric monetary system is to fully separate both parts by using two monies of equal
supply and circulating the asset units of one and the liability units of the other. The
respective complementary parts are kept at a central bank.
In an analogy to physics, the asset units can be considered as money whereas liability units
are antimoney [1, 40]. In this case, the analogy however is limited in so far as money and
antimoney do not annihilate. Demand deposits at commercial banks are to be fully backed
by central bank money, i.e. commercial banks would be prohibited from credit creation. Of
course, time and investment deposits could still be fractionally backed. In order to be able to
adapt the money supply to economic growth, the central bank would be allowed to directly
10
regulate the amount of money in circulation.
Now agent i can either purchase some good from agent j by transfering ∆a asset units
to him or by receiving ∆l liability units along with the good. To put it briefly, in order
for agents to increase their monetary wealth, they want to obtain asset units and get rid of
liability units. The feasibility and pitfalls of such a system will be discussed in section VIII.
Of course, ∆a and ∆l do not have to be the same. The introduction of antimoney allows the
market to continuously assess the value of assets and liabilities and to establish two different
nominal price levels pa and pl for both payment modes. As with any two currencies, this
gives rise to a free floating exchange rate between both monies that is missing in the present
system. Let us define the equilibrium exchange rate as eeq =
rate in a trade as e =
∆a
.
∆l
pa
pl
and the implicit exchange
Real monetary wealth in a symmetric monetary system is given
by
ω=
a
l
− ,
p a pl
(10)
where a and l denote the asset and liability holdings. This is the linear extension to the
definition of real purchasing power in economic textbooks [4]. Now if a transfer is executed
by either of the two payment options and e =
∆a
∆l
= eeq , the monetary wealth of the payee
changes by
∆ω =
∆a
e∆l
=
.
pa
pa
(11)
If the seller offers both payment options but asks for prices such that e 6= eeq , he is asking
for a premium for one of the payment options. For example he might ask for a premium if
the buyer “pays in liabilities” (i.e. if the buyer agrees to receive liability units):
∆a
∆l
<
.
eeq · pl
e · pl
(12)
In addition to the new payment option via liabilities there is now a novel, interesting way
to provide liquidity to an agent by simultaneously transferring asset and liability units to
him. Assume for a moment that both price levels are equal: pa = pl . Then an agent can
transfer ∆a = ∆l asset and liability units to a liquidity-seeking market participant without
changing his monetary wealth. Thus, in a symmetric monetary system as outlined above,
monetary wealth and liquidity become different notions, as illustrated in Figure 4.
Let us turn again to the general case of arbitrary pa and pl . If a liquidity provider P
simultaneously transfers δa asset units and δl liability units, his monetary wealth changes
11
FIG. 4. The liquidity of an agent can be limited by either (a) its asset or (b) its liability holdings.
The blue line indicates the amount of money and antimoney that would be transferred at a liquidity
price φ. Here we have assumed φ > 1, i.e. a liquidity provider P would choose to transfer more
liability units than asset units. His profit then depends on the equilibrium exchange rate eeq .
by
∆ωP = −
δa δl
+ .
pa
pl
(13)
If he wants to be remunerated for his services, he will set a nominal price of liquidity
φ=
δl
δa
(14)
such that ∆ωP > 0. Assume for example an investor who requires means of payments
equivalent to a wealth ∆ω =
δa
.
pa
Then the provider’s wealth changes by
δa δa · φ
δa
∆ωP = − +
=
pa
pl
pa
pa
φ−1
pl
= (eeq φ − 1) ∆ω.
(15)
Thus, in order to profit from the provision of liquidity, the provider should ask for a price
φ>
1
.
eeq
The formal definition of liquidity of an agent i can then be stated as
li
Li = min ai ,
.
φ
(16)
How could such a concept help to overcome the issue of credit crunches commonly associated with full reserve banking systems? In a conventional system with 100% reserve
requirements, a borrower has to find someone who is actually willing to part with his money
for a certain time. This is not the case in a fractional reserve banking system in which the
12
commercial bank can accommodate the loan. In a system of money and antimoney, a buyer
would be able to accept antimoney instead of paying with money. Thus, if he wasn’t able
to find the necessary “money funds”, a trade could still succeed if the seller had sufficient
antimoney.
Another advantage lies in the nature of the liquidity market. In contrast to a system
where liquidity is provided by additional loan contracts, the price of liquidity is set at the
moment of transfer. In a conventional monetary system the price is determined by the
payment of interest, often fixed for a long time span, which carries with it the need for
supervision and enforcing the contractual agreements. Thus the legal overhead including
time delays is non-negligible for a single agent. In a symmetric monetary system, however,
the provision of liquidity is as simple as an ordinary act of purchase.
V.
TIME-HOMOGENEOUS MONEY IN RANDOM ECONOMIES
In order to make predictions about economic quantities in a monetary system, we need a
model that describes the interactions between market participants. As human behaviour is
notoriously hard, if not impossible, to capture in models, a considerable amount of research
in this field has focused on random economies in which agents exchange random amounts
of money, much like particles exchange energy [2, 41–43]. This reflects the fact that the
environment and the future of investments are hard to predict also for the agents — in the
extreme case the environment is considered fully random. Surprisingly, such an economic null
model correctly predicts monetary wealth distributions for all but the richest subpopulation
[41]. Moreover, it is a challenging test bed to probe the stability of monetary systems [2, 44].
We study the time-homogeneous monetary system proposed in the previous section with
the model of a random economy. Agents exchange random amounts xi of asset and liability
units drawn from an exponential price distribution
p(xi ) =
1 − xpi
e i,
pi
(17)
where i ∈ {a, l} and pi is the respective price level. That particular choice is motivated by
private observations [1] which confirm that in real economies transfers with low prices are
encountered much more often than high prices. However, the distribution of transfer prices
is not critical to the outcome: Dragulescu and Yakovenko have shown that the equilibrium
13
monetary distribution in a closed single currency system is given by a Boltzmann-Gibbs distribution for a wide range of random transfer schemes, namely those that have time-reversal
symmetry [43]. If we consider the money and antimoney holdings to be independent of each
other (i.e. if we disregard the liquidity providing mechanism introduced in the previous section), we should expect both distributions to equilibrate to Boltzmann-Gibbs distributions
as well:
Pi (x) =
where i ∈ {a, l}. The parameter Ti =

x

 1 · e− Ti
Ti
if x ≥ 0

0
if x < 0,
Mi
,
N
(18)
with Mi being the total amount of asset or liability
units and N being the number of market participants, is equivalent to a temperature in
statistical physics. The corresponding simulation in Figure 5 (a) confirms this expectation.
In contrast to a single currency economy, the wealth distribution is now distinct from the
money distribution. Let P (ω, a, l) be the probability of finding an agent with wealth ω, a
assets and l liabilities. Considering that the probability distributions of a and l are only
well-defined for positive values, the marginal probability of finding an agent with wealth ω
is:
ZZ
P (ω) =
dadlP (ω, a, l)
ZZ
a
l
=
dadlδ ω −
−
P (a)P (l)
pa pl
Z∞
pa l
= |pa | ·
dlP pa ω +
P (l).
pl
(19)
max (−wpl ,0)
Inserting equation 18 for Pa and Pl , we find that for positive values of ω the distribution is
given by
Z∞
1
pa l
l
dl exp −
pa ω +
· exp −
Ta
pl
Tl
0
pa pl
ωpa
=
· exp −
.
Tl pa + Ta pl
Ta
|pa |
·
P (ω) =
Ta · Tl
14
(20)
FIG. 5. Money and wealth distributions with (c,d) and without (a,b) a liquidity market (LM).
(a) In a random economy without liquidity transfer, asset and liability holdings both equilibrate to
a Boltzmann-Gibbs distribution. (b) According to equation 19, the monetary wealth distribution
also depends on the price levels pa and pl and may thus be asymmetric. The black lines in (b) were
calculated using equations 20 and 21 whereas points represent random transfer simulations with
N = 100000 agents. (c,d) Simulation results with a liquidity market. (c) Except for two outliers
in the upper left corner, the distribution of assets and liabilities still largely follows an exponential
law with effective temperatures T 6=
M
N.
(d) If a liquidity market is present (shown for φ = 2 and
φ = 5), the resulting wealth distribution will always be broader than in the case of no liquidity
trading. With large φ, the wealth distribution becomes increasingly asymmetric. The simulation
was performed using pa = pl and Ta = Tl .
15
FIG. 6. Monetary wealth distribution with debt caps ω̂ on individual debt. The underlying asset
distribution remains unchanged, but the liability distribution changes its temperature due to the
cutoff. As credit creation is prohibited in a symmetric system, the global debt is restricted to the
total amount of antimoney. The simulation was performed with N = 100000 agents, T = 10 and
pa = pl = 20.
For negative values of ω one finds
Z∞
1
pa l
l
dl exp −
pa ω +
· exp −
Ta
pl
Tl
−ωpl
pa pl
ωpl
=
· exp
.
Tl pa + Ta pl
Tl
|pa |
P (ω) =
·
Ta · Tl
(21)
So we see that the wealth in a symmetric monetary system with a random economy is
distributed according to a Laplace distribution (see also Figure 5 (b)). Interestingly, this
distribution is very similar to the one found in a system with fractional reserve banking [42].
This is to be expected since both impose a cap on global dept by limiting the total number
of asset-liability pairs. The first moment of the wealth is given by
2
pa pl
Ta
Tl2
hωi =
·
− 2 .
Tl pa + Ta pl
p2a
pl
(22)
In order to prevent agents from accumulating an unlimited amount of antimoney, it could
seem sensible to additionally impose a cap on individual debt. In that case, the wealth
distribution is cut off at some ω̂. The resulting wealth distributions are shown in Figure
6. As will be discussed later, there are other ways to prevent the hoarding of antimoney.
The percentage of creditors in a symmetric system without individual debt caps, i.e. the
16
FIG. 7. Fraction of agents f (∆m) who can pay ∆m or more. Note that for prices lower than
∆m = 2T ln 2, a symmetric monetary system with a total money supply M = T · N is advantageous
even to a single currency system with M 0 = 2 · M . In this case we chose T = 10 and have assumed
pa = pl . The black lines are calculated according to equations 25, 27 and 26, whereas points
represent simulation results with N = 100000 agents.
percentage of agents with ω > 0 is
Z∞
Pω>0 =
dωP (ω)
0
=
1
1 + ppal TTal
(23)
and by the same reasoning the number of debtors (ω < 0) is
Z0
Pω<0 =
dωP (ω)
−∞
=
1
.
1 + pplaTTal
(24)
Of course, we find that Pω>0 + Pω<0 = 1.
VI.
PROBABILITY OF SUCCESSFUL TRADES
We would like to analyze the trading success of the agents. Even within a random model
economy, this quantity can give hints on whether or not a monetary system is prone to credit
crunches. Let us have a look on the fraction f1 of agents who can pay an amount ∆m in a
17
single currency economy:
Z∞
f1 (∆m) =
p(x)dx
∆m
∆m
.
= exp −
T
(25)
In the time-homogeneous system, a trade that would fail if carried out via asset units could
still succeed through the payment with liabilities. If we assume pa = pl = 1 for simplicity,
the fraction of agents who can pay ∆m is
f3 (∆m) = 2e−
∆m
T
.
(26)
Contrary to what one might expect, this differs from a doubling of the money supply to
T 0 = 2 · T in a single currency system in which case we would have
Z∞
f2 (∆m) =
p0 (x)dx
∆m
∆m
.
= exp −
2·T
(27)
The graphs of f1 , f2 and f3 are depicted in Figure 7 along with simulation results. For prices
lower than ∆m = 2T ln 2, the time-homogeneous monetary system – even without trading
liquidity – increases the chances for a successful trade compared to a single currency system
with T 0 = 2 · T .
In a closed single currency economy with exponentially distributed prices, the number of
successful trades can be calculated quite easily. With p∗ being the price level and T being
the temperature of the single currency, we get
∗
Z∞
f (p ) =
0
∆
e− p∗
p∗
Z∞
m
e− T
T
dmd∆ =
.
T
T + p∗
(28)
∆
Figure 8 (a) shows that this fits the simulation data very well. In a time-homogeneous,
symmetric system without a liquidity market, the calculation can still be done analytically.
Let P (S, ∆a , ∆l , m) be the joint probability that the transfer prices in a trade are ∆a and ∆l ,
respectively, that the trade is successful and that transfer mode m is chosen. There are three
different transfer modes: either the trade is carried out via asset-units or via liability-units
or the trade fails. We denote these events by A, L and F . By marginalizing, we find the
fraction of successful trades:
18
P (S) =
X ZZ
d∆a d∆l P (S, ∆a , ∆l , m)
m
=
X ZZ
ZmZ
=
d∆a d∆l P (S|∆a , ∆l , m)P (m|∆a , ∆l )P (∆a , ∆l )
h
d∆a d∆l P (S|∆a , ∆l , A)P (A|∆a , ∆l )P (∆a , ∆l )
P (S|∆a , ∆l , L)P (L|∆a , ∆l )P (∆a , ∆l )
i
P (S|∆a , ∆l , F )P (F |∆a , ∆l )P (∆a , ∆l )
ZZ
h
i
=
d∆a d∆l P (A|∆a , ∆l ) + P (L|∆a , ∆l ) P (∆a , ∆l )
!
" Z∞
m
Z∞
ZZ
−m
e Tl
e− Ta
· dm
dm
=
d∆a d∆l
Ta
Tl
∆l
∆a
Z∞
+
Z∞
m
e− Ta
dm
·
Ta
1−
∆a
1−
m
e− Ta
dm
Ta
=
l
!
Tl
! Z∞
#
−m
e Tl
· dm
· P (∆a , ∆l )
Tl
∆a
ZZ
m
−T
∆l
Z∞
+
dm
e
∆l
h ∆a
∆
− l
d∆a d∆l e− Ta · e Tl
∆ ∆a
− l
+ e− Ta · 1 − e Tl
∆ i
∆a
− l
+ 1 − e− Ta · e Tl · P (∆a , ∆l ).
(29)
If both ∆a and ∆l are independently drawn from equation 17, i.e. if
P (∆a , ∆l ) = P (∆a ) · P (∆l ),
(30)
the fraction of successful trades in the economy would be
f (pa , pl ) = P (S) =
Ta
Tl
Ta · Tl
+
−
.
pa + Ta pl + Tl (pa + Ta ) · (pl + Tl )
(31)
A more realistic scenario would be that prices are connected through an exchange rate e as
introduced in the third section:
∆a
P (∆a , ∆l ) = δ ∆l −
e
In that case, the fraction of successful trades is
19
· P (∆a ).
(32)
FIG. 8. (a) Fraction of successful trades in a closed single currency random economy and in a
time-homogeneous, symmetric system, both with exponentially distributed transfer prices. The
black lines are calculated according to equations 28 and 33. As one would expect, higher price
levels lead to less successful trades since agents simply do not possess enough means of payment to
carry out the trades. Due to the additional payment mode, a system of money and antimoney leads
to a higher number of successful trades. With the liquidity market switched on, the fraction of
successful trades increases even further as agents can now easily obtain means of payment. (b) The
fraction of successful trades in a symmetric, time-homogeneous system without a liquidity market
for two different price distributions P (∆a , ∆l ). The points represent simulation data and the black
lines are calculated according to equations 31 and 33. Of course, in all cases the fraction of failing
trades is f¯ = 1 − f . The simulation was performed with 10000 agents and Ta = Tl = 8.
f (pa , e) =
eTl
Ta · eTl
Ta
+
−
.
pa + Ta pa + eTl pa · eTl + pa Ta + Ta · eTl
(33)
Both cases are depicted in Figure 8 (b) together with the corresponding simulation results
which confirm the analytic solution. In the next section we will see that the introduction of
a liquidity market leads to a significantly increased number of successful trades.
VII.
THE LIQUIDITY MARKET
So far we did not make use of the liquidity providing mechanism possible in a timehomogeneous, symmetric system and impossible in a single currency system. Now we switch
on the liquidity market such that agents who possess liquidity according to equation 16 will
20
provide it to agents whose trades would otherwise fail because neither the asset holdings of
the buyer nor the liability holdings of the seller are sufficient to execute the transfer.
The specific implementation is such that we keep a list of potential liquidity providers
and their liquidity. If a trade can neither be executed via money nor antimoney, the buyer
will randomly ask agents on that list until the trade can either be realised or there are no
more liquidity providers left. In the latter case the trade ultimately fails. We start with the
most simple assumption that the price φ of liquidity is exogenously given in the simulation.
Before we turn to the results of the simulation, let us first calculate the probability of finding
an agent who could provide liquidity if the liquidity market is switched off:
Z∞
dy pa (x) · pl (y)
dx
PL =
0
=
Z∞
φx
1
1+φ·
Ta
Tl
.
(34)
As can be seen from Figure 9, this describes the simulation results very well. If we now
allow liquidity trading, the amount of “free” liquidity decreases sharply as it is all used up
in trades.
Switching on the liquidity market also changes the money and wealth distribution. When
comparing Figure 5 (c,d) with Figure 5 (a,b), one can see that the resulting distributions with
a liquidity market can still be described by exponential functions except for two ’outliers’
in the upper left corner. The outliers represent the liquidity providers who are left with
little asset and liability holdings and the buyers of initially failing trades who had to turn
to the liquidity market. They end up with little asset but larger liability holdings. For the
remaining agents we see that the effective temperatures of the fitted exponential distributions
(see equation 18) have changed compared to Figure 5 (a,b). Note that this does not mean
that
M
N
has changed.
Figure 10 (a) shows the effective temperatures of assets and liabilities for various prices
of liquidity φ. In Figure 10 (b) we have plotted three asset distributions to illustrate the
effect. For the liability distributions one would find the reverse effect, i.e. the slopes get
shallower with increasing φ. As can be seen from Figure 5 (d), the wealth distribution is
broadened if liquidity trading is enabled. For larger values of φ, the wealth distribution
becomes increasingly asymmetric, seen in the divergence of the effective temperatures Ta
and Tl (see Figure 10). This makes sense, since buyers of liquidity have to accept more
21
FIG. 9. The amount of free liquidity in a time-homogeneous, symmetric random economy with
and without a liquidity market. Both (a) and (b) show the same data with different scalings for
the y-axis. The simulation was performed using 10000 agents Ta = Tl = 100 and pa = pl = 200.
With increasing φ, insufficient liability holdings become the limiting factor for an agent’s liquidity
according to equation 16. A liquidity market leads to a rapid redistribution of excess liquidity. Note
that for pa = pl , prices φ < 1 would probably not be encountered in a real economy as this would
constitute a loss of monetary wealth for the liquidity provider. The smaller φ, the longer it takes
to reach low free liquidity levels. This effect can still be seen for the data point that corresponds
to φ = 0.01. The black lines were calculated according to equation 34.
liabilities than assets for φ > 1. A similar asymmetry was found in Figure 5 (b) for an
exchange rate e =
pa
pl
=
1
3
< 1. Naturally, the price of liquidity φ =
δl
δa
also reflects the price
levels and thus φ and the exchange rate e are expected to be inversely proportional.
As a final observation, we see from Figure 8 (a) that the fraction of successful trades is
increased when a liquidity market is present. This is to be expected as agents who initially
lack liquidity can now turn to the liquidity market and obtain means of payment. Due to the
spot trade property, i.e. the absence of long-term contracts (see section IV), this mechanism
should be able to alleviate the problem of credit crunches in real systems with 100% reserve
requirements.
22
FIG. 10. (a) Effective temperatures for different liquidity prices. φ = 0 is the case of no liquidity
market. Note that even for high values of φ, the asset and liability distributions will always be
broader than in the case of no liquidity trading. Asset and liability temperatures diverge with
increasing φ as the wealth distributions in Figure 5 (d) become increasingly asymmetric. In (b) we
have plotted the asset distribution for three different liquidity prices φ for better illustration of the
phenomenon. Note that for increasing φ the slope becomes steeper which corresponds to a lower
temperature. For the liability distribution one finds the reverse effect (only shown in (a)).
VIII.
DISCUSSION
A real implementation of a time-homogeneous, symmetric monetary system has only
recently become conceivable with the advent of electronic payment systems. Since liability
units or antimoney constitute an obligation to render goods or services to society, agents
will want to get rid of it. It is obvious that such a system will require a way to make the
individual antimoney holdings transparent. One would have to make sure that no agent
exclusively consumes by accumulating antimoney without ever passing it on to other agents,
i.e. without contributing to society. Money on the other hand would not need monitoring.
For the liquidity market to be efficient, money should preferably be as easily transferable as
antimoney.
The history of debt is laden with ways of how creditors enforce the payback of loans.
An overview from anthropology was recently provided [6]. One rather inelegant possible
solution, already discussed in Figure 6, would be to impose a debt cap, i.e. transfers would
only be allowed if the payer’s wealth did not drop below a certain threshold ω̂. Another
23
solution would be to assign time-stamps to antimoney which would determine the period
within which the antimoney had to be passed along. This could also be achieved by an
electronic implementation.
The question is whether a market mechanism alone could prevent agents from hoarding
antimoney. Closely related to this point is the issue of bankruptcy and defaults. It turns out
that the legal framework is likely to be similar to today’s rules. If an agent had acquired some
good by receiving antimoney and went bankrupt afterwards, i.e. he would be unable to pass
on the antimoney within the set time window, the seller had to take the antimoney back and
would in turn become responsible again for passing it on. This is the equivalent of a write-off
today. Thus, the negotiation of antimoney prices would depend on the buyer’s economic
standing. For large transaction volumes, financial service providers could provide insurance
against an agent’s default. Similar to contemporary regulations which implement rules for
personal default, an insurance against leaving the system with improper and imbalanced
money and antimoney holdings could be implemented. We expect that the exchange rate
dynamics between money and antimoney allow other modes of handling liabilities which are
not yet conceived from our traditional monetary background using credit and interest.
How would the market change the prices of antimoney with respect to the prices in money
units? With decreasing antimoney prices, it is more likely that sellers will choose payments in
money. The wealth of the agents that have exposed themselves to large antimoney holdings
would decrease according to equation 10. Without further experimental analysis, we can
only speculate whether the exchange rate between money and antimoney together with the
liquidity market could establish a stable market equilibrium by itself. Both the historical
record [6] and the most recent credit crises have shown at length that a creditary economy
is not providing society a stable market equilibrium.
In a time-homogeneous, symmetric system, money creation by commercial banks is prohibited. However, it might well be sensible to peg the amount of central bank money to
certain economic quantities. For a conventional system, Irving Fisher suggested to choose
the amount of central bank money proportional to the total number of market participants
[33], an approach that we implemented in our simulations. Other regulatory measures such
as Friedman’s k-percent rule should also not be ruled out a priori [45, 46]. It states that
the central bank should increase the money supply every year by a constant rate that is
irrespective of economic cycles.
24
IX.
CONCLUSION
We have introduced a new bi-currency system that allows exploiting the advantages of full
reserve banking systems without giving up easy access to liquidity. We tested the approach
within the economic scheme of random transfers and could provide a number of analytical
solutions within that framework. Due to the symmetry of the system it is possible for
agents to provide liquidity to other market participants without the need for overseeing
and enforcing long-term credit contracts on a personal level. This mechanism is new over
systems with only one kind of money and facilitates the transfer of excess liquidity.
ACKNOWLEDGMENTS
The authors would like to thank Axel Schenzle and Ulrich Schollwöck for discussions and
Ludwig Ohl and Robert Fischer for discussions and providing feedback on the manuscript
at various stages.
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